Computer Assisted Proof for Normally Hyperbolic Invariant Manifolds

We present a topological proof of the existence of a normally hyperbolic invariant manifold for maps. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. But a non-rigorous, good enough, guess is necessary. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method for a driven logistic map, for which non-rigorous numerical simulation in plain double precision suggests the existence of a chaotic attractor. We prove that this numerical evidence is false and that the attractor is a normally hyperbolic invariant curve.Comment: 33 pages, 16 figure

Transition Tori in the Planar Restricted Elliptic Three Body Problem

We consider the elliptic three body problem as a perturbation of the circular problem. We show that for sufficiently small eccentricities of the elliptic problem, and for energies sufficiently close to the energy of the libration point L2, a Cantor set of Lyapounov orbits survives the perturbation. The orbits are perturbed to quasi-periodic invariant tori. We show that for a certain family of masses of the primaries, for such tori we have transversal intersections of stable and unstable manifolds, which lead to chaotic dynamics involving diffusion over a short range of energy levels. Some parts of our argument are nonrigorous, but are strongly backed by numerical computations

Cone Conditions and Covering Relations for Topologically Normally Hyperbolic Invariant Manifolds

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. We provide conditions which imply the existence of the manifold within an investigated region of the phase space. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method to obtain an invariant manifold within an explicit range of parameters for the rotating H\'enon map

Derivative pricing methodology in continuous-time models

AbstractWe show that the fundamental methodology (and practice) of evaluation of derivative securities in continuous-time models is consistent with discrete-time theory, in which a derivative price is based on the principle that adding this security to the market does not create a violation of the basic economic principle: no riskless profit with zero investment

Thermal Conductivity of Isotopically Enriched 28Si Revisited

The thermal conductivity of isotopically enriched 28Si (enrichment better than 99.9%) was redetermined independently in three laboratories by high precision experiments on a total of 4 samples of different shape and degree of isotope enrichment in the range from 5 to 300 K with particular emphasis on the range near room temperature. The results obtained in the different laboratories are in good agreement with each other. They indicate that at room temperature the thermal conductivity of isotopically enriched 28Si exceeds the thermal conductivity of Si with a natural, unmodified isotope mixture by 102 %. This finding is in disagreement with an earlier report by Ruf et al. At 26 K the thermal conductivity of 28Si reaches a maximum. The maximum value depends on sample shape and the degree of isotope enrichment and exceeds the thermal conductivity of natural Si by a factor of 8 for a 99.982% 28Si enriched sample. The thermal conductivity of Si with natural isotope composition is consistently found to be 3% lower than the values recommended in the literature